# Questions tagged [cauchy-product]

For questions about the Cauchy product, the discrete convolution of two sequences.

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### Multiplying two polynomials: explanation of the general formula for the coefficients

If $$f(x)=a_nx^n+...+a_0$$ and $$g(x)=b_mx^m+…+b_0$$ then $$f(x)\cdot g(x)=c_{m+n}x^{m+n}+...+c_0$$ where $c_k=\sum_{r+s=k}a_rb_s,\quad k=0,....,m+n$ I know that the degree $0$ and $(m+n)$ exists, ...
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### Find the Cauchy product of $e^x$ and $e^{-x}$.

I was able to get to: $c_n$ = $$\sum_{k=0}^{n} \binom{n}{k} \frac{x^k (-x)^{n-k}}{n!}$$ but I am stuck as to how to get past this. Is there a property of the series that lets me pull the negative ...
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### Unclear equation in my scriptum (perhaps cauchy-product) [duplicate]

Hi the following equation in my scriptum seems unclear. I think it has something to do with cauchy-product but i dont know
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### Power Series Solution for an ODE which has trigonometric coefficient functions

The ODE for which we seek a power series solution is: $$y''+ \cos(x)y' + x\sin(x)y = 0,\hspace{0.4cm} y(0) = 1,\hspace{.1cm} y'(0) = 0$$ I need to find the partial sum up to five, from the initial ...
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### Cauchy Product starting from $1$

The definition of the Cauchy product from Wikipedia is defined as $$\left(\sum_{i=0}^\infty a_i\right)\left(\sum_{j=0}^\infty b_j\right) =\sum_{k=0}^\infty\sum_{\ell=0}^ka_\ell b_{k-\ell}$$ ...
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### Find limit of sum

I suspect that $\lim_{n \to \infty} \sum_{k = 0}^{n - 1}\frac{k}{a^k(n - k)} = 0$ for $a > 1$. I know that this product represents the taylor coefficients of $\frac{-ax\ln(1 - x)}{(a - x)^2}$ by ...
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### The series $\sum_{k=1}^\infty\sum_{n=1}^k k^{-3}/(2n-1)$

$$\sum_{k=1}^\infty\sum_{n=1}^k\frac{1}{(2n-1)k^3}$$ Can anyone help me find this series? I tried to use Cauchy product but I don't know how I can complete it.
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### Convergent Cauchy product of divergent series

I was looking at the counterexample $a_n = b_n = (-1)^n/\sqrt{n}$ where $\sum a_n$ and $\sum b_n$ converge but the Cauchy product $\sum_{k=0}^\infty c_k = \sum_{k=0}^\infty \sum_{j=0}^{k}a_j b_{k-j}$ ...
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### Cauchy product of $\sum\limits_n^{\infty}\frac{1}{n}$ with itself [closed]
Is the cauchy product of $\sum\limits_n^{\infty}\frac{1}{n}$ with itself simply $\sum\limits_n^{\infty}\frac{1}{n^2}$? I can't apply the definition here https://proofwiki.org/wiki/Definition:...